Efficient digital filter design tool for approximating an FIR filter with a low-order linear-phase IIR filter

ABSTRACT

A method and apparatus for designing low-order linear-phase IIR filters is disclosed. Given an FIR filter, the method utilizes a new Krylov subspace projection method, called the rational Arnoldi method with adaptive orders, to synthesize an approximated IIR filter with small orders. The method is efficient in terms of computational complexity. The synthesized IIR filter can truly reflect essential dynamical features of the original FIR filter and indeed satisfies the design specifications. In particular, the linear-phase property is stilled remained in the passband.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method of designing digital filters,and more particularly, to a method of designing low-order linear-phaseIIR filters for approximating an FIR filter.

2. Description of Related Art

Digital filters for addressing digital signal processing have beenapplied widely in commercial electronic products, such as compact diskplayers, television sets and the like. In order to deal with real-timesignals, the designs should be considered low computational costs tooutput signals efficiently. Also, since the distortion-free transmissionof-the waveforms in the passband is very important in signal processing,the filters should contain a linear-phase characteristic, i.e., constantgroup delay.

There are two classes of digital filters. One is a finite-durationimpulse response (FIR) filters and the other is an infinite-durationimpulse response (IIR) filters. The main advantages of the FIR filtersare exactly linear phase and guaranteed stable. However, when thespecification being very rigorous, the resulting FIR filter is usuallywith higher orders, which may require more hardware components and lowerthe operational speed. Conversely, the IIR filters are useful forlarge-scale or high-speed designs but they do not have exactly linearphase and can not guarantee to be stable.

One way to synthesize the IIR filters with linear phase in passband isto solve the rational approximation problem directly. These methods, forexample, include Pade approximation, linear programming, nonlinearprogramming, multiple criterion optimization and eigenfilter approach.

Another way is called the indirect approach. It will be composed ofthree steps, as shown in FIG. 1. Step 1 receives and stores the designspecifications of a digital filter.

These specifications are required in the frequency-domain in terms ofthe desired magnitude and phase response of the filter. Then, alinear-phase FIR filter, which meets design specifications, will bedesigned in step 2. The order and the coefficients of the FIR filter canbe obtained using the conventional methods such as thefrequency-sampling design technique, the window design technique and theoptimal equiripple design technique. Finally, a lower-order IIR filterwill be obtained using filter approximation techniques in step 3. Itshould be mentioned that the special attentions shall be paid on thisindirect approach. The resulting IIR filter must be ensured to capturethe linear-phase response of the original FIR filter in the passband.

In recent years, several linear-phase IIR filter design techniques havebeen emerged for this purpose. Generally speaking, two distinct methodshave be proposed: (1) Grammian-Based Methods: including the balancedtruncation method and impulse response grammian method and (2) OptimalApproximation Methods: including the least-square approximations and theH₂ norm approximation. Although satisfactory results have been reported,computational complexity of these methods are still quite expensive.

SUMMARY OF THE INVENTION

The main objective of the present invention is to provide awater-preventing grommet for a pull chain switch, which efficientlykeeps water out of the pull chain switch to avoid malfunction.

The secondary objective of the present invention is to provide awater-preventing grommet for a pull chain switch, which smoothesoperations of the pull chain switch.

To achieve the objectives, the method of approximating an FIR filterwith low-order linear-phase IIR filters by the rational Arnoldialgorithm with adaptive orders in accordance with the present inventioncontains the following steps:

-   -   (a) initialize the first vector of the Krylov sequence for each        expansion point;    -   (b) in the jth iteration of the algorithm, choose an expansion        frequency such that the frequency gives the greatest difference        between the (j+1)st-order output moment of the original FIR        filter H(z) and that of the lower-order IIR filter Ĥ(z);    -   (c) after the chosen expansion point in jth iteration being        determined, the single-point Arnoldi method applied at the        expansion point to generate the new orthnormal vector;    -   (d) determine the new residual at each expansion point for the        next iteration; wherein the resulting orthogonal projection        matrix is output after giving the total iteration number of the        algorithm.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an indirect approach to design a low-orderlinear-phase IIR filter;

FIG. 2 illustrates the design flow of the low-order linear-phase IIRfilters;

FIG. 3 illustrates the typical design specifications of a low-passfilter;

FIG. 4 shows the detail flow of the rational Arnoldi method withadaptive orders; and

FIGS. 5A-7C show the bode plots of the magnitude, the error inmagnitude, and the phase of the original FIR filters and the low-orderIIR filters.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 2 shows the design flow of a low-order linear-phase IIR filter inthe present invention in which includes the steps of receiving andstoring the design specifications in step 1, designing an FIR filtersatisfying the design specifications and saving the order andcoefficients in step 2, establishing the state space matrices {A,b,c} instep 3, performing the rational Arnoldi method with adaptive orders andproduce the orthogonal projection matrix V in step 4, and generating thecorresponding low-order linear-phase IIR filter, which can approximatethe original FIR filter and satisfy the design specifications in step 5.

FIG. 3 illustrates typical design specifications of a low-pass filter instep 1, where the band [0,ω_(p)] (unit 34) is called the passband and δ₁(unit 36) is the acceptable tolerance (or ripple) in the passband, theband [ω_(s),π] (unit 38) is called the stopband and δ₂ (unit 40) is thecorresponding tolerance (or ripple), and the band [ω_(p),ω_(s)] (unit42) is called the transition band. R_(p) is the passband ripple in dB,where R_(p)=−20 log₁₀ [(1−δ₁)/(1+δ₁)]. A_(s) is the stopband attenuationin dB, where A_(s)=−20 log₁₀ [δ₂/(1+δ₁)]. Notably, either {δ₁,δ₂} or{R_(p), A_(s)} is required to be stored in step 1.

Suppose that an FIR filter has been designed to satisfy the designspecifications in step 2. Let H(z)=Σ_(i=0) ^(n)h_(i)z^(−i) be the causalFIR filter with length n+1. A state-space realization of H(z) in step 3can be described as

$\begin{matrix}{\begin{matrix}{{x( {k + 1} )} = {{{Ax}(k)} + {{bu}(k)}}} \\{{y(k)} = {{c^{T}\;{x(k)}} + {h_{0}{u(k)}}}}\end{matrix},} & (1) \\{where} & \; \\\begin{matrix}{{A = \begin{bmatrix}0 & 0 & \cdots & 0 & 0 \\1 & 0 & \cdots & 0 & 0 \\0 & 1 & \cdots & 0 & 0 \\\vdots & \vdots & \vdots & \vdots & \vdots \\0 & 0 & \cdots & 1 & 0\end{bmatrix}},} & {{b = \begin{bmatrix}1 \\0 \\\vdots \\\vdots \\0\end{bmatrix}},} & {{c = \begin{bmatrix}h_{1} \\h_{2} \\\vdots \\\vdots \\h_{n}\end{bmatrix}},}\end{matrix} & (2)\end{matrix}$and AεR^(n×n), bεR^(n), cεR^(n). The transfer function H(z) can also beexpressed as H(z)=c^(T)X(z)+h₀=c^(T)(zI_(n)−A)⁻¹b+h₀. Our problemformulation is to find a lower-order IIR filter Ĥ(z), which satisfiesthe same specifications in step 1 as the original FIR filter H(z) andmaintains a linear-phase response in the passband.

The way in the invention is to find an optimal IIR filter by usingorthogonal projection of the original FIR filter. By matching somecharacteristics of the original FIR filter, the resulting orthonormalmatrix V can be generated in step 4. The lower-order IIR filter Ĥ(z) canbe constructed using the orthonormal projection x(k)=V{circumflex over(x)}(k). In such a situation, the parameters of the IIR filter can bedefined by the following congruence transformation in step 5,Â=V^(T)AV, {circumflex over (b)}=V^(T)b, and ĉ=V^(T)c.  (3)It can be shown that the matrix V^(T) AV is always stable as long as (1)and matrix A is stable, (2) V^(T)V=I. Thus, the stability of thelower-order IIR filter generated by Eq. (3) is guaranteed.Pade Approximation and Moment Matching

The basis theory of the method in the invention is the multi-point Padeapproximation, or so called the multi-point moment matching, to obtain alow-order IIR filter. Expanding X(z) in power series about variousfrequencies {z₁, z₂, . . . , z_(î)}, where each z_(i)=e^(jω) ^(i) εC and0≦ω_(i)≦π, we have

$\begin{matrix}{{{X(z)} = {\sum\limits_{j = 0}^{\infty}{{X^{(j)}( z_{i} )}( {z - z_{i}} )^{j}}}},} & (4) \\{where} & \; \\{{{X^{(j)}( z_{i} )} = {\lbrack {- ( {{z_{i}I_{n}} - A} )^{- 1}} \rbrack^{j}( {{z_{i}I_{n}} - A} )^{- 1}b}},} & (5) \\\begin{matrix}{{H^{(j)}( z_{i} )} = {c^{T}\;{X^{(j)}( z_{i} )}}} & \; & {( {j > 0} ),}\end{matrix} & \; \\{{H^{(0)}( z_{i} )} = {{c^{T}\;{X^{(0)}( z_{i} )}} + {h_{0}.}}} & \;\end{matrix}$X^((j))(z_(i)) is called the jth-order system moment of X(z);H^((j))(z_(i)) represents the jth-order output moment of H(z) at z_(i).Notably, if î=1, Eq. (4) is indeed the conventional Pade approximation.The objective is to find a q-order (q<n) IIR filterĤ(z)=ĉ^(T)(zI_(q)−Â)⁻¹{circumflex over (b)}+h₀ such thatH^((j))(z_(i))=Ĥ^((j))(z_(i)) for j=0,1, . . . , ĵ_(i)−1 and i=1,2, . .. , î, where q=Σ_(i=1) ^(î)ĵ_(i).

It shall be mentioned that moment calculations can be obtainedanalytically by exploring special characteristics of matrices A and b ineq. (2). For each z_(i), (z_(i)I_(n)−A)⁻¹b and (z_(i)I_(n)−A)⁻¹ can bederived analytically as the following formulas:

$\begin{matrix}{{( {{z_{i}I_{n}} - A} )^{- 1} = \begin{bmatrix}{1/z_{i}} & 0 & \cdots & 0 & 0 \\{1/z_{i}^{2}} & {1/z_{i}} & \cdots & 0 & 0 \\{1/z_{i}^{3}} & {1/z_{i}^{2}} & \cdots & 0 & 0 \\\vdots & \vdots & \vdots & \vdots & \vdots \\{1/z_{i}^{n - 1}} & {1/z_{i}^{n - 2}} & \cdots & {1/z_{i}} & 0 \\{1/z_{i}^{n}} & {1/z_{i}^{n - 1}} & \cdots & {1/z_{i}^{2}} & {1/z_{i}}\end{bmatrix}},} \\{{( {{z_{i}I_{n}} - A} )^{- 1}b} = {\begin{bmatrix}{1/z_{i}} & {1/z_{i}^{2}} & \cdots & {1/z_{i}^{n}}\end{bmatrix}^{T}.}}\end{matrix}$Krylov Subspace and the Arnoldi Method

Explicitly computing moments usually yields numerically ill-conditionedproblems. We adapt recent works about the Krylov space method to solvethese problems. Given a square matrix ΨεC^(n×n) and a vector ξεC^(n),the qth Krylov sequenceK _(q)(Ψ, ξ)≡span(ξ,Ψξ,Ψ²ξ, . . . ,Ψ^(q−1)ξ)is a sequence of q column vectors and the corresponding column space iscalled the qth Krylov subspace. Set Ψ=(z_(i)I_(n)−A)⁻¹ andξ=(z_(i)I_(n)−A)⁻¹b. It has been shown that the Krylov subspaceK_(q)(Ψ,ξ) is indeed spanned by the system moments X^((j))(z_(i)) forj=0,1, . . . , q−1. The Arnoldi method, a kind of Krylov subspacemethods, is employed to generate an orthonormal matrix V_(q) that spansthe same subspace as the Krylov subspace K_(q)(Ψ,ξ). As a result, theguaranteed stable IIR filter can be constructed by substituting V_(q)into Eq. (3).

The Arnoldi method arises from the Hessenberg reduction A=VHV^(T) foreigenvalue calculations. It has the advantage that it can be terminatedpart-way and leaving one with a partial reduction to a Hessenberg form.The process is exploited to form iterative algorithms. During theiteration process, an upper Hessenberg matrix H_(q)εC^(q×q) is generatedthat satisfies the following relationship:ΨV _(q) =V _(q) H _(q) +h _(q+1,q) v _(q+1) e _(q) ^(T) and v₁=ξ/∥ξ∥,  (6)where e_(q) is the qth unit vector in R^(q). The vector v_(q+1)satisfies a (q+1)-term recurrence relation, involving itself and thepreceding Krylov vectors. A new orthonormal vector v_(q+1) can begenerated using the modified Gram-Schmidt orthogonalization technique.The Rational Arnoldi Method

Generally speaking, the accuracy of the Pade approximation based methodsis lost away from the expansion point more rapidly as the eigenvalues ofthe FIR filter approach the expansion frequency. A rational Arnoldi (RA)method, which uses multiple expansion points, was developed to overcomethis difficulty. The straightforward way for multi-point moment matchingapplications is to apply the Krylov subspace algorithm at variousexpansion frequencies. This is the so-called rational Krylov algorithm.Basically, this algorithm is a generalization of theshifted-and-inverted Arnoldi algorithm. To simplify the developments,the number of the matched moments of the lower-order IIR filter at eachexpansion point is assumed to be fixed. Formally, let Z={z₁, z₂, . . . ,z_(î)} represent the set of predetermined expansion frequencies. LetJ={ĵ₁, ĵ₂, . . . , ĵ_(î)} be the set of the number of the matchedmoments at each corresponding frequency. The rational Arnoldi methodwill generate a lower-order IIR filter Ĥ(z), which matches q-order(q=Σ_(i=1) ^(î)ĵ_(i)) moments of the FIR filter, H(z), at the expansionpoints z_(i), i=1,2, . . . , î.

Implementing the rational Arnoldi method is equivalent to implement theArnoldi method ĵ_(i) times at î expansion frequencies. That is, thefirst ĵ₁ iterations correspond to the expansion frequency z₁ and thenext ĵ₂ iterations are associated with z₂, and so on. Each Arnoldiiteration generates ĵ_(i) orthonormal vectors. Then, V_(q)=└v₁ v₂ . . .v_(q)┘ is the desired orthonormal matrix generated from a union Krylovspace at various expansion points, as stated byK _(q)=span(X ⁽⁰⁾(z ₁), . . . ,X ^((ĵ) ¹ ⁻¹⁾(z ₁), . . . ,X ⁽⁰⁾(z _(î)),. . . ,X ^((ĵ) ^(î) ⁻¹⁾(z _(î))).

Once the orthonormal matrix V_(q) has been formed by applying therational Arnoldi method and the lower-order IIR filter can be obtainedusing the congruence transformation.

The Rational Arnoldi Method with Adaptive Orders

Selecting a set of expansion points z_(i) for i=1,2, . . . , î and thenumber of matched moments ĵ_(i) about each z_(i) is by no means trivial.For simplicity, the expansion points z_(i) for i=1,2, . . . , î aredetermined in advance using engineering heuristics or experimentalmeasurements over a specified frequency range. This invention describesan intelligent scheme for choosing multiple expansion points in each ofthe iterations.

Suppose that H^((j))(z_(i))=Ĥ^((j))(z_(i)) for j=0,1, . . . , ĵ_(i)−1and i=1,2, . . . , î after q iterations of the rational Arnoldialgorithm. However, the ĵ_(i)th-order output moments H^((ĵ) ^(i)⁾(z_(i))=Ĥ^((ĵ) ^(i) ⁾(z_(i)) can not be guaranteed. The concept thatunderlies the rational Arnodli method with adaptive orders is to selectan expansion point z_(i*) _(q+1) in the (q+1)st iteration. Hence, theresulting (q+1)st-order IIR filter yields the greatest momentimprovement |H^((ĵ) ^(i) ⁾(z_(i))−Ĥ^((ĵ) ^(i) ⁾(z_(i))| of the qth-orderIIR filter as z_(i)=z_(i*) _(q+1) . The moment errors can be directlyobtained in the new iteration without explicitly calculating systemmoments.

FIG. 4 displays the detail flow of the rational Arnoldi method withadaptive orders in step 4 in FIG. 2.

Step 1, in FIG. 4, initializes the first vectork⁽⁰⁾(z_(i))=(z_(i)I_(n)−A)⁻¹b of the Krylov sequence for each expansionpoint z_(i), where iε{1, . . . , î}. Since the lower-order IIR filterand the orthonormal matrix are not yet determined, the residuer⁽⁰⁾(z_(i)) for each z_(i) is set to k⁽⁰⁾(z_(i)). The normalizationcoefficient about each z_(i), h_(π)(z_(i)), is initialized to be one.Step 2, in FIG. 4, begins the iterations and sets j=1.

Step 3, in FIG. 4, chooses an expansion frequency z_(i) such that z_(i)gives the greatest difference between the (j+1)st-order output moment ofthe original FIR filter H(z) and that of the lower-order IIR filterĤ(z), that is,max_(z) _(i) _(εZ) |H ^((j+1))(z _(i))−Ĥ ^((j+1))(z _(i))|=max_(z) _(i)_(εZ) |h _(π)(z _(i))c ^(T) r ^((j−1))(z _(i))|.Ĥ^((j+1))(z_(i)) is the (j+1)st-order output moment of the lower-orderIIR filter Ĥ(z), which is yielded using the congruence transformationmatrix V_(j−1) (j>1) and matches j-order output moments of H(z) atz_(i). The chosen expansion frequency in the jth iteration is calledz_(i*) _(j) .

After choosing the expansion point z_(i*) _(j) in the determined jthiteration, the single-point Arnoldi method is applied at the expansionpoint z_(i*) _(j) (unit 52), which contains steps 4 and 5, as shown inFIG. 4. Step 4, in FIG. 4, generates the new orthnormal vector v_(j) andthe vector is incorporated into the orthnormal matrix V_(j−1). Thenormalization coefficient h_(π)(z_(i))=Π_(j)∥r^((j−1))(z_(i))∥ whenz_(i) is selected in the jth iteration.

Step 5, in FIG. 4, determines the new residual r^((j))(z_(i)) at eachexpansion point z_(i). The calculation involves a projection with thenew orthonormal matrix V_(j). The next vector k^((j))(z_(i*) _(q+1) ) atthe frequency z_(i*) _(q+1) must be updated to enable further matchingof the output moment in the (j+1)st iteration. Since no improvement isobtained at the other unselected frequency z_(i), the vectork^((j))(z_(i)) at frequency z_(i) in the current iteration remainsk^((j−1))(z_(i)), which was obtained in the preceding iteration. Resetj=j+1 in step 6 and judge if j≦q in step 7, as shown in FIG. 4. Finally,the resulting orthogonal projection matrix V_(q) is generated in step 8in FIG. 4.

The resulting orthnormal matrix V_(q) should be real to ensure that realsystem matrices of the lower-order IIR filter are generated if thecomplex expansion frequencies are used. First, all column vectors inV_(q) are divided into the real part V_(r) and the imaginary part V_(i).Second, a reduced QR factorization of [V_(r) V_(i)] is performed toyield a new orthogonal matrix V_(q). The moment matching property of theresulting lower-order IIR filter by the new and real V_(q) is alsopreserved.

The details of the algorithm are outlined as follows. The vector Zincludes î expansion points, q is the total number of iterations andV_(q) is the resulting orthonormal matrix.

Adaptive Rational Arnoldi (input: A,b,c,Z,q; output: V_(q)) (1): /*Initialize */ 1 for each z_(i) ∈ Z do 2   k⁽⁰⁾(z_(i)) :=(z_(i)I_(n)−A)⁻¹b, r⁽⁰⁾(z_(i)) := k⁽⁰⁾(z_(i)) 3   h_(π)(z_(i)) := 1 4end for (2): /* Begin the Iterations */ 5 for j = 1, 2, . . ., q do    (2.1) /* Select the Expansion Frequency with the Maximum        OutputMoment Error*/ 6   Choose z_(i) ∈ Z as the i givingmax_(i)|h_(π)(z_(i))c^(T)r^((j−1))(z_(i))| 7   set z_(i*) _(j) be theexpansion frequency in the jth iteration     (2.2) /*Generate theOrthonormal Vector at z_(i*) _(j) */ 8   h_(j,j−1)(z_(i*) _(j) ) :=||r^((j−1))(z_(i*) _(j) )|| 9   v_(j) = r^((j−1))(z_(i*) _(j))/h_(j,j−1)(z_(i*) _(j) ) 10   h_(π)(z_(i*) _(j) ) := h_(π)(z_(i*) _(j))·h_(j,j−1)(z_(i*) _(j) )     (2.3) /* Update the Residue r^((j))(z_(i))for the Next Iteration */ 11    for each z_(i) ∈ Z do 12     if (z_(i)== z_(i*) _(j) ) then k^((j))(z_(i*) _(j) ) := −(z_(i)I_(n)−A)⁻¹v_(j) 13    else k^((j))(z_(i)) := k^((j−1))(z_(i)) 14     end if 15    r^((j))(z_(i)) := k^((j))(z_(i)) 16       for t = 1, 2, . . ., j do17         h_(t,j)(z_(i)) := v_(t) ^(H)r^((j))(z_(i)) 18        r^((j))(z_(i)) := r^((j))(z_(i))−h_(t,j)(z_(i))v_(t) 19      end for 20    end for 21 end for 22 V_(q) = [V₁ V₂ . . . V_(q)]

Some properties of the method of approximating an FIR filter bylow-order IIR filters in the invention are summarized as follows.

(1) Exact expression of output moment errors: suppose that the outputmoments of the original FIR filter and those of the lower-order IIRfilter are matched, that is, H^((j))(z_(i))=Ĥ^((j))(z_(i)) for j=0,1, .. . , ĵ_(i)−1 and i=1,2, . . . , î. The system matrices of thelower-order IIR filter are generated by the congruence transformationwith the orthonormal matrix V_(q) using the algorithm, where q=Σ_(i=1)^(î)ĵ_(i). The magnitude error between the ĵ_(i)th-order moments H^((ĵ)^(i) ⁾(z_(i)) and Ĥ^((ĵ) ^(i) ⁾(z_(i)) at each expansion point z_(i) canbe expressed as follows:|H ^((ĵ) ^(i) ⁾(z _(i))−Ĥ ^((ĵ) ^(i) ⁾(z _(i))|=|h _(π) c ^(T) r ^((ĵ)^(i) ⁻¹⁾(z _(i))|,  (10)where h_(π)(z_(i))=Π_(j)∥r^((j−1))(z_(i))∥.

(2) Moment matching can still be preserved.

(3) In the first iteration in the rational Arnodli algorithm withadaptive orders, step (2.2) is to choose z_(i)εZ such thatmax(|c^(T)(z_(i)I_(n)−A)⁻¹b|)=max(|H(z_(i))|). This is equivalent tofind out the expansion frequency with the maximum magnitude in theoutput frequency response.

(4) Implementation issues of digital filters: the present invention alsoprovides several heuristics of selecting expansion frequencies inadvance for the proposed rational Arnoldi method. Generally speaking,the complex expansion points {z₁, z₂, . . . , z_(î)} will berecommended, where each z_(i)=e^(jω) ^(i) εC and 0≦ω_(i)≦π. Then thefrequency responses of the lower-order IIR filters at these points canbe the same as those of the original FIR filter. Nevertheless, if realexpansion points can be selected, the computational complexity ofyielding approximate IIR filters can be further reduced. The followingguidelines are provided:

(a) Low-pass/high-pass filters: the proposed method with the expansionpoint ω₁=0 performs well over the low frequency range of responses. Forhigh-pass filter designs, the special structures of state-space matricesmay be used to present the duality between low-pass and high-passfilters. Let Ā=−A, b=b, c=c, and h₀ =−h₀,

${\overset{\_}{H}(z)} = {{{{{\overset{\_}{c}}^{T}( {{zI}_{n} - \overset{\_}{A}} )}^{- 1}\overset{\_}{b}} + {\overset{\_}{h}}_{0}} = {\sum\limits_{i = 0}^{n}{( {- 1} )^{i + 1}h_{i}{z^{- i}.}}}}$

If H(z) presents a high-pass filter, then H(z) will be a low-passfilter, and a vice versa. Likewise, the expansion point ω₁=0 is chosento perform the Arnoldi algorithm. If the corresponding orthonormalmatrix V _(q) is obtained, then the high-pass IIR filter, whichsatisfies the same specifications as the original FIR filter, can beconstructed as follows:Â= V _(q) ^(T)A V _(q), {circumflex over (b)}= V _(q) ^(T)b, and ĉ= V_(q) ^(T)c.

(b) Band-pass/band-stop filters: experimental results indicate that thepassband edge and stopband edge frequencies are appropriate candidateexpansion points in meeting the specifications of the design. Otherexpansion points with uniform spacing are also recommend to be selected.

DESIGN EXAMPLES

Three example filters are used to justify the proposed approach. Table 1describes specifications of a low-pass filter, a high-pass filter, and aband-pass filter. The command remez in Matlab was used to design the FIRfilters by the optimal equiripple technique. Table 2 lists thecorresponding orders. Then, the approximate low-order IIR filters weregenerated by the proposed method and the balanced realization method(BAL). Table 2 shows the reduced orders and the expansion points used bythe two methods. FIGS. 5A-7C display the bode plots of the magnitude,the error in magnitude, and the phase of the original FIR filters andthe low-order IIR filters. In FIGS. 5A-7C, the responses of the originalFIR filters are represented as thin solid lines. Those of the IIRfilters, determined by the proposed method, are represented as thicksolid lines -, and those determined by BAL method are plotted as thickdashed lines - -. The responses in the passband of the IIR filters areindistinguishable from those of the original FIR filters, independentlyof which the model reduction method is used. Simulation results implythat the performance of the proposed method is similar to that of theBAL method in the passband. The resulting lower-order IIR filters canactually preserve the linear-phase response of the original FIR filters.Nevertheless, in terms of computational efficiency, the Kylov subspacebased methods generally outperform the BAL method.

TABLE 1 Filter design specifications Specifications Low-Pass High-PassBand-Pass Maximum passband attenuation 3 1 1 (dB) Maximum stopbandattenuation 40 35 35 (dB) Lower passband edge (rad/s) 0 0.85π 0.3255πUpper passband edge (rad/s) 0.285π   1π 0.3755π Lower stopband edge(rad/s) 0.353π 0 0.6655π Upper stopband edge (rad/s)    1π 0.78π 0.7155π

TABLE 2 Matched orders and expansion points for IIR filter designsLow-Pass High-Pass Band-Pass FIR Order 41 41 58 IIR Order 24 17 36Expansion ω = 0 ω = {0.4π, 0.6π} points

CONCLUSIONS

A rational Arnoldi method with adaptive orders for approximating FIRfilters by low-order linear-phase IIR filters has been proposed. Thedeveloped method is very efficient in terms of computational complexity.Meanwhile, the lower-order IIR filter can truly reflect the dynamicalfeatures of the FIR filter and satisfies the original designspecifications.

Although the invention has been explained in relation to its preferredembodiment, it is to be understood that many other possiblemodifications and variations can be made without departing from thespirit and scope of the invention as hereinafter claimed.

1. A method of approximating an FIR filter with low-order linear-phaseIIR filters by the rational Arnoldi algorithm with adaptive orderscontaining the following steps: a) initialize the first vector of theKrylov sequence for each expansion point; b) in the jth iteration of thealgorithm, choosing an expansion frequency wherein the heuristics ofselecting expansion frequencies in advance for the proposed rationalArnoldi method we given by (a) low-pass filters: the proposed methodwith the expansion point ω₁=0; (b) high-pass filters: the specialstructures of state-space matrices used to present the duality betweenlow-pass and light pass filters; let state matrices become Ā=−A, b=b,c=c, and h₀ =−h₀, the expansion point ω₁=0 chosen to perform the Arnoldialgorithm; when the corresponding orthonormal matrix V _(q) is obtainedand then the high-pass IIR filter, which satisfies the samespecifications as the original FIR filter; and (c) band-pass/band-stopfilters: the passband edge and the stopband edge frequencies being theappropriate candidate expansion points in meeting the specifications ofthe design, and other expansion points with uniform spacing recommendedto be selected such that the frequency gives the greatest differencebetween the (j+1)st-order output moment of the original FIR filter H(z)and that of the lower-order IIR filter Ĥ(z) wherein the expression ofoutput moment errors between the ĵ_(i)th-order moments H^((ĵ) ^(i)⁾(z_(i)) and Ĥ^((ĵ) ^(i) ⁾(z_(i)) at each expansion point z_(i) areexpressed as follows:H^((ĵ_(i)))(z_(i)) − Ĥ^((ĵ_(i)))(z_(i)) = h_(π)c^(T)r^((ĵ_(i) − 1))(z_(i)), where  $\;{{h_{\pi}( z_{i} )} = {\prod\limits_{j}\;{{{r^{({j - 1})}( z_{i} )}}}}}$is the normalization coefficient when an expansion frequency z_(i) isselected in the jth iteration; vector c contains the last n impulseresponse coefficients of a FIR filter with length n+1; andr^((j−1))(z_(i)) is the residual vector in the (j−1)st iteration of thedisclosed adaptive rational Arnoldi algorithm at the expansion frequencyz_(i); c) after the choosing the expansion point in jth iteration beingdetermined, the single-point Arnoldi method applied at the expansionpoint to generate the new orthnormal vector; and d) determine a newresidual at each expansion point for next iteration; whereby, after thegiving total iteration number of the algorithm, outputting the resultingorthogonal projection matrix.